![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rspsn | Structured version Visualization version GIF version |
Description: Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
rspsn.b | ⊢ 𝐵 = (Base‘𝑅) |
rspsn.k | ⊢ 𝐾 = (RSpan‘𝑅) |
rspsn.d | ⊢ ∥ = (∥r‘𝑅) |
Ref | Expression |
---|---|
rspsn | ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2744 | . . . . 5 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ (𝑎(.r‘𝑅)𝐺) = 𝑥) | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
3 | 2 | rexbidv 3173 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
4 | rlmlmod 20626 | . . . 4 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
5 | rlmsca2 20622 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
6 | baseid 17045 | . . . . . 6 ⊢ Base = Slot (Base‘ndx) | |
7 | rspsn.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 6, 7 | strfvi 17021 | . . . . 5 ⊢ 𝐵 = (Base‘( I ‘𝑅)) |
9 | rlmbas 20616 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
10 | 7, 9 | eqtri 2765 | . . . . 5 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
11 | rlmvsca 20623 | . . . . 5 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
12 | rspsn.k | . . . . . 6 ⊢ 𝐾 = (RSpan‘𝑅) | |
13 | rspval 20614 | . . . . . 6 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
14 | 12, 13 | eqtri 2765 | . . . . 5 ⊢ 𝐾 = (LSpan‘(ringLMod‘𝑅)) |
15 | 5, 8, 10, 11, 14 | lspsnel 20416 | . . . 4 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ ∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺))) |
16 | 4, 15 | sylan 580 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ ∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺))) |
17 | rspsn.d | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
18 | eqid 2737 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | 7, 17, 18 | dvdsr2 20028 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (𝐺 ∥ 𝑥 ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
20 | 19 | adantl 482 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐺 ∥ 𝑥 ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
21 | 3, 16, 20 | 3bitr4d 310 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ 𝐺 ∥ 𝑥)) |
22 | 21 | abbi2dv 2876 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2714 ∃wrex 3071 {csn 4584 class class class wbr 5103 I cid 5528 ‘cfv 6493 (class class class)co 7351 ndxcnx 17024 Basecbs 17042 .rcmulr 17093 Ringcrg 19917 ∥rcdsr 20019 LModclmod 20274 LSpanclspn 20384 ringLModcrglmod 20582 RSpancrsp 20584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-sca 17108 df-vsca 17109 df-ip 17110 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 df-minusg 18711 df-sbg 18712 df-subg 18883 df-mgp 19855 df-ur 19872 df-ring 19919 df-dvdsr 20022 df-subrg 20172 df-lmod 20276 df-lss 20345 df-lsp 20385 df-sra 20585 df-rgmod 20586 df-rsp 20588 |
This theorem is referenced by: lidldvgen 20677 zndvds 20908 |
Copyright terms: Public domain | W3C validator |